1,058 research outputs found

    Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs

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    The study of graph products is a major research topic and typically concerns the term f(GH)f(G*H), e.g., to show that f(GH)=f(G)f(H)f(G*H)=f(G)f(H). In this paper, we study graph products in a non-standard form f(R[GH]f(R[G*H] where RR is a "reduction", a transformation of any graph into an instance of an intended optimization problem. We resolve some open problems as applications. (1) A tight n1ϵn^{1-\epsilon}-approximation hardness for the minimum consistent deterministic finite automaton (DFA) problem, where nn is the sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this implies the hardness of properly learning DFAs assuming NPRPNP\neq RP (the weakest possible assumption). (2) A tight n1/2ϵn^{1/2-\epsilon} hardness for the edge-disjoint paths (EDP) problem on directed acyclic graphs (DAGs), where nn denotes the number of vertices. (3) A tight hardness of packing vertex-disjoint kk-cycles for large kk. (4) An alternative (and perhaps simpler) proof for the hardness of properly learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004 and J. Comput.Syst.Sci. 2008]

    On Routing Disjoint Paths in Bounded Treewidth Graphs

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    We study the problem of routing on disjoint paths in bounded treewidth graphs with both edge and node capacities. The input consists of a capacitated graph GG and a collection of kk source-destination pairs M={(s1,t1),,(sk,tk)}\mathcal{M} = \{(s_1, t_1), \dots, (s_k, t_k)\}. The goal is to maximize the number of pairs that can be routed subject to the capacities in the graph. A routing of a subset M\mathcal{M}' of the pairs is a collection P\mathcal{P} of paths such that, for each pair (si,ti)M(s_i, t_i) \in \mathcal{M}', there is a path in P\mathcal{P} connecting sis_i to tit_i. In the Maximum Edge Disjoint Paths (MaxEDP) problem, the graph GG has capacities cap(e)\mathrm{cap}(e) on the edges and a routing P\mathcal{P} is feasible if each edge ee is in at most cap(e)\mathrm{cap}(e) of the paths of P\mathcal{P}. The Maximum Node Disjoint Paths (MaxNDP) problem is the node-capacitated counterpart of MaxEDP. In this paper we obtain an O(r3)O(r^3) approximation for MaxEDP on graphs of treewidth at most rr and a matching approximation for MaxNDP on graphs of pathwidth at most rr. Our results build on and significantly improve the work by Chekuri et al. [ICALP 2013] who obtained an O(r3r)O(r \cdot 3^r) approximation for MaxEDP

    Computation-Aware Data Aggregation

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    Data aggregation is a fundamental primitive in distributed computing wherein a network computes a function of every nodes\u27 input. However, while compute time is non-negligible in modern systems, standard models of distributed computing do not take compute time into account. Rather, most distributed models of computation only explicitly consider communication time. In this paper, we introduce a model of distributed computation that considers both computation and communication so as to give a theoretical treatment of data aggregation. We study both the structure of and how to compute the fastest data aggregation schedule in this model. As our first result, we give a polynomial-time algorithm that computes the optimal schedule when the input network is a complete graph. Moreover, since one may want to aggregate data over a pre-existing network, we also study data aggregation scheduling on arbitrary graphs. We demonstrate that this problem on arbitrary graphs is hard to approximate within a multiplicative 1.5 factor. Finally, we give an O(log n ? log(OPT/t_m))-approximation algorithm for this problem on arbitrary graphs, where n is the number of nodes and OPT is the length of the optimal schedule

    Maximum Edge-Disjoint Paths in kk-sums of Graphs

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    We consider the approximability of the maximum edge-disjoint paths problem (MEDP) in undirected graphs, and in particular, the integrality gap of the natural multicommodity flow based relaxation for it. The integrality gap is known to be Ω(n)\Omega(\sqrt{n}) even for planar graphs due to a simple topological obstruction and a major focus, following earlier work, has been understanding the gap if some constant congestion is allowed. In this context, it is natural to ask for which classes of graphs does a constant-factor constant-congestion property hold. It is easy to deduce that for given constant bounds on the approximation and congestion, the class of "nice" graphs is nor-closed. Is the converse true? Does every proper minor-closed family of graphs exhibit a constant factor, constant congestion bound relative to the LP relaxation? We conjecture that the answer is yes. One stumbling block has been that such bounds were not known for bounded treewidth graphs (or even treewidth 3). In this paper we give a polytime algorithm which takes a fractional routing solution in a graph of bounded treewidth and is able to integrally route a constant fraction of the LP solution's value. Note that we do not incur any edge congestion. Previously this was not known even for series parallel graphs which have treewidth 2. The algorithm is based on a more general argument that applies to kk-sums of graphs in some graph family, as long as the graph family has a constant factor, constant congestion bound. We then use this to show that such bounds hold for the class of kk-sums of bounded genus graphs

    Non-approximability and Polylogarithmic Approximations of the Single-Sink Unsplittable and Confluent Dynamic Flow Problems

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    Dynamic Flows were introduced by Ford and Fulkerson in 1958 to model flows over time. They define edge capacities to be the total amount of flow that can enter an edge in one time unit. Each edge also has a length, representing the time needed to traverse it. Dynamic Flows have been used to model many problems including traffic congestion, hop-routing of packets and evacuation protocols in buildings. While the basic problem of moving the maximal amount of supplies from sources to sinks is polynomial time solvable, natural minor modifications can make it NP-hard. One such modification is that flows be confluent, i.e., all flows leaving a vertex must leave along the same edge. This corresponds to natural conditions in, e.g., evacuation planning and hop routing. We investigate the single-sink Confluent Quickest Flow problem. The input is a graph with edge capacities and lengths, sources with supplies and a sink. The problem is to find a confluent flow minimizing the time required to send supplies to the sink. Our main results include: a) Logarithmic Non-Approximability: Directed Confluent Quickest Flows cannot be approximated in polynomial time with an O(log n) approximation factor, unless P=NP. b) Polylogarithmic Bicriteria Approximations: Polynomial time (O(log^8 n), O(log^2 kappa)) bicritera approximation algorithms for the Confluent Quickest Flow problem where kappa is the number of sinks, in both directed and undirected graphs. Corresponding results are also developed for the Confluent Maximum Flow over time problem. The techniques developed also improve recent approximation algorithms for static confluent flows

    Dynamic vs Oblivious Routing in Network Design

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    Consider the robust network design problem of finding a minimum cost network with enough capacity to route all traffic demand matrices in a given polytope. We investigate the impact of different routing models in this robust setting: in particular, we compare \emph{oblivious} routing, where the routing between each terminal pair must be fixed in advance, to \emph{dynamic} routing, where routings may depend arbitrarily on the current demand. Our main result is a construction that shows that the optimal cost of such a network based on oblivious routing (fractional or integral) may be a factor of \BigOmega(\log{n}) more than the cost required when using dynamic routing. This is true even in the important special case of the asymmetric hose model. This answers a question in \cite{chekurisurvey07}, and is tight up to constant factors. Our proof technique builds on a connection between expander graphs and robust design for single-sink traffic patterns \cite{ChekuriHardness07}

    Routing with Congestion in Acyclic Digraphs

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    We study the version of the kk-disjoint paths problem where kk demand pairs (s1,t1)(s_1,t_1), \dots, (sk,tk)(s_k,t_k) are specified in the input and the paths in the solution are allowed to intersect, but such that no vertex is on more than cc paths. We show that on directed acyclic graphs the problem is solvable in time nO(d)n^{O(d)} if we allow congestion kdk-d for kk paths. Furthermore, we show that, under a suitable complexity theoretic assumption, the problem cannot be solved in time f(k)no(d/logd)f(k)n^{o(d/\log d)} for any computable function ff
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